Multi Rate DSP
Multi Rate DSP
Multi Rate DSP
Processing
Multirate Digital Signal
Processing
x[ n ] xa ( nT ) M y[ n ] xa ( nMT )
x[ n ] xa ( nT ) L y[n]
x ( nT / L ), n 0, L, 2 L,
a
0 otherwise
Frequency-Domain Characterization
• Consider first a factor-of-2 up-sampler
whose input-output relation in the time-
domain is given by
x[n / 2], n 0, 2, 4,
x u [n ]
0, otherwise
Up-Sampler
y[n] x[Mn]
we get
n
Y ( z) x[Mn] z
n
• The expression on the right-hand side cannot
be directly expressed in terms of X(z)
Down-Sampler
x[n ] H (z M ) M y1 [ n ]
Cascade equivalence #2
x[n ] L H (z L ) y2 [ n]
x[n ] H (z ) L y2 [ n]
Filters in Sampling Rate
Alteration Systems
• The bandwidth of a critically sampled
signal must be reduced by lowpass
filtering before its sampling rate is reduced
by a down-sampler to avoid aliasing
• Likewise, the zero-valued samples
introduced by an up-sampler must be
interpolated by lowpass filtering to more
appropriate values for an effective
sampling rate increase
Filter Specifications
• Since up-sampling causes periodic
repetition of the basic spectrum, the
unwanted images in the spectra of the up-
sampled signal xu [n] must be removed by
using a lowpass filter H(z), called the
interpolation filter,
filter as indicated below
xu [n]
x[n ] L H (z) y[n ]
j 1, c / M
H (e )
0, / M
• The design of the filter H(z) is a standard
IIR or FIR lowpass filter design problem
Polyphase Decomposition
• The FIR filter is realized using direct form
• To avoid unnecessary calculations the decimator
is replaced with efficient transversal structure.
N 1
y (n) h[m]xm [n]
m 0
= z0[... + h(0) z0 + h(M ) z−M + ...] + z−1[... + h(1) + h(M + 1) z−M + ...]
+ z−2[... + h(2) + h(M + 2) z−M + ...] + ...
+ z−(M −1) [... + h(M − 1) + h(2M − 1) z−M + ...]
M −1 +∞
H (z) = ∑ z P (z ) ∑ z h(nM + i)
−i M −n
⇒ i where Pi (z) =
i=0 n=−∞
Polyphase Decomposition
Implementation of Decimation
Using noble identity: